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Fractional Grönwall--Wendroff Inequalities for Implicit Systems with Distributed Memory

Rômulo Damasclin Chaves dos Santos

TL;DR

This paper develops a unified analytical framework for implicit fractional differential systems with distributed memory and delays by deriving multivariate Grönwall–Wendroff-type inequalities tailored to weakly singular kernels. It proves existence and uniqueness via a fixed-point approach in weighted Banach spaces and establishes Ulam–Hyers stability with explicit bounds, enabling robust analysis under perturbations. The framework is applied to the delayed fractional FitzHugh–Nagumo model, yielding rigorous conditions for the existence of nontrivial periodic solutions (limit cycles) and illuminating how the fractional order $\alpha$ and delay $\tau$ jointly influence neural excitability. The results provide explicit parameter–dependence laws (e.g., $I_{\mathrm{th}} \propto \tau^{1-\alpha}$ as $\tau\to0$) and offer analytical tools for a broad class of memory-driven, implicit dynamical systems across physics, biology, and engineering.

Abstract

This work establishes a comprehensive analytical framework for studying implicit fractional differential systems with distributed memory and time delays. We develop novel fractional integral inequalities of Grönwall--Wendroff type that are specifically adapted to handle multivariate functions with singular kernels and implicit dependencies. These inequalities provide essential a priori estimates for analyzing complex memory-dependent systems. Building upon these results, we prove general existence and uniqueness theorems for implicit fractional differential equations using fixed-point theory in appropriately weighted Banach spaces. Furthermore, we establish Ulam--Hyers stability criteria, demonstrating that small perturbations in the governing equations lead to proportionally small deviations in solutions. The theoretical advances are applied to the fractional FitzHugh--Nagumo model with delay (FHN-$α$-$τ$), a neurodynamical system exhibiting both subdiffusive memory and discrete time delays. Our analysis yields rigorous conditions for the existence of limit cycles corresponding to action potentials, along with asymptotic stability criteria. The results reveal how fractional order $α$ and delay $τ$ jointly modulate neural excitability thresholds and dynamic regime transitions. This work bridges fundamental fractional calculus theory with applications in mathematical neuroscience, providing analytical tools for systems where present evolution depends non-trivially on historical states and intrinsic fractional rates of change.

Fractional Grönwall--Wendroff Inequalities for Implicit Systems with Distributed Memory

TL;DR

This paper develops a unified analytical framework for implicit fractional differential systems with distributed memory and delays by deriving multivariate Grönwall–Wendroff-type inequalities tailored to weakly singular kernels. It proves existence and uniqueness via a fixed-point approach in weighted Banach spaces and establishes Ulam–Hyers stability with explicit bounds, enabling robust analysis under perturbations. The framework is applied to the delayed fractional FitzHugh–Nagumo model, yielding rigorous conditions for the existence of nontrivial periodic solutions (limit cycles) and illuminating how the fractional order and delay jointly influence neural excitability. The results provide explicit parameter–dependence laws (e.g., as ) and offer analytical tools for a broad class of memory-driven, implicit dynamical systems across physics, biology, and engineering.

Abstract

This work establishes a comprehensive analytical framework for studying implicit fractional differential systems with distributed memory and time delays. We develop novel fractional integral inequalities of Grönwall--Wendroff type that are specifically adapted to handle multivariate functions with singular kernels and implicit dependencies. These inequalities provide essential a priori estimates for analyzing complex memory-dependent systems. Building upon these results, we prove general existence and uniqueness theorems for implicit fractional differential equations using fixed-point theory in appropriately weighted Banach spaces. Furthermore, we establish Ulam--Hyers stability criteria, demonstrating that small perturbations in the governing equations lead to proportionally small deviations in solutions. The theoretical advances are applied to the fractional FitzHugh--Nagumo model with delay (FHN--), a neurodynamical system exhibiting both subdiffusive memory and discrete time delays. Our analysis yields rigorous conditions for the existence of limit cycles corresponding to action potentials, along with asymptotic stability criteria. The results reveal how fractional order and delay jointly modulate neural excitability thresholds and dynamic regime transitions. This work bridges fundamental fractional calculus theory with applications in mathematical neuroscience, providing analytical tools for systems where present evolution depends non-trivially on historical states and intrinsic fractional rates of change.
Paper Structure (7 sections, 6 theorems, 45 equations)

This paper contains 7 sections, 6 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Fractional Wendroff-Type Integral Inequalities
  3. Existence and Uniqueness for Implicit Fractional Systems
  4. Ulam--Hyers Stability
  5. Application to the FHN-$\alpha$-$\tau$ Model
  6. Main Results
  7. Conclusion

Key Result

Theorem 2.2

Let $u,f \in C([0,T],\mathbb{R}^n)$ with $u(t)\ge 0$ and $f(t)\ge 0$ componentwise. Assume that for all $t\in[0,T]$, where $\alpha,\beta\in(0,1]$, $A,B\in L^\infty([0,T],\mathbb{R}^{n\times n})$, and $u(t)=\phi(t)$ for $t\in[-\tau,0]$. Then there exists a constant $M>0$, depending only on $\alpha,\beta,\|A\|_\infty,\|B\|_\infty$ and $T$, such that

Theorems & Definitions (15)

  • Definition 2.1
  • Theorem 2.2: Multivariate Fractional Grönwall--Wendroff Inequality
  • proof
  • Theorem 3.1: Existence and Uniqueness
  • proof
  • Definition 4.1
  • Theorem 4.2: Ulam--Hyers Stability
  • proof
  • Lemma 5.1: Fractional dissipativity
  • proof
  • ...and 5 more